Finite Group Databasebeta

← #30669275

Subgroups — #30669275

Order 16768.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523512C211yesmaximal, characteristic, complemented
2C2×#16862214C212yesmaximal, complemented
3C2×C22⋊C4×D131C211yesmaximal, characteristic, complemented
4#22553225C211yesmaximal, characteristic, complemented
5#22553224C211yesmaximal, characteristic, complemented
6#22553220C211yesmaximal, characteristic, complemented
7#22553192C212yesmaximal, complemented
8#22553189C212yesmaximal, complemented
9#22553188C212yesmaximal, complemented
10#22553174C211yesmaximal, characteristic, complemented
11#392×C131C211yesmaximal, characteristic, complemented
12C24×D131C2211yescharacteristic
13C2×#14523512—28no—
14C2×#14523512—24no—
15C22×Dic262C2211yescharacteristic, complemented
16C22×C22⋊D131C2212yes—
17C22×C22⋊D131C2212yes—
18C23×Dic131C2212yescomplemented
19C22×C4×D131C2211yescharacteristic
20C22×C4×D131C2211yescharacteristic
21#16862214—28no—
22C2×C22⋊Dic131C2211yescharacteristic
23C2×C22⋊Dic131C2212yes—
24C2×C22⋊Dic131C2212yescomplemented
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2211yescharacteristic
27C2×(C2×D131)⋊C4C2212yes—
28C2×(C2×D131)⋊C4C2211yescharacteristic
29C22⋊C4×D131—24no—
30C131⋊#95C2212yescomplemented
31C131⋊#95C2211yescharacteristic, complemented
32C131⋊#95C2212yescomplemented
33C131⋊#95C2212yescomplemented
34C22×D4×C131C2212yes—
35C22×D4×C131C2211yescharacteristic, complemented
36C2×C22⋊C4×C131C2211yescharacteristic
37C2×C22⋊C4×C131C2212yescomplemented
38#95×C131C2211yescharacteristic, complemented
39C23×D131—22no—
40C23×D131—22no—
41C23×D131—24no—
42C23×D131—22no—
43C23×D131—42no—
44C23×D131C2311yescharacteristic
45C23×D131C2311yescharacteristic
46#14523512—232no—
47C2×Dic262—24no—
48C2×Dic262—22no—
49C2×C22⋊D131—24no—
50C2×C22⋊D131—24no—
51C2×C22⋊D131—24no—
52C2×C22⋊D131—24no—
53C2×C22⋊D131—42no—
54C2×C22⋊D131D418yescomplemented
55C22×Dic131—24no—
56C22×Dic131—24no—
57C22×Dic131—24no—
58C22×Dic131C2311yescharacteristic
59C22×Dic131C2312yes—
60C22×Dic131C2311yescharacteristic
61C22×Dic131C2312yes—
62C2×C4×D131—24no—
63C2×C4×D131—22no—
64C2×C4×D131—42no—
65C2×C4×D131D414yescomplemented
66C22⋊Dic131—24no—
67C22⋊Dic131—22no—
68C22⋊Dic131—42no—
69(C2×D131)⋊C4—24no—
70(C2×D131)⋊C4—24no—
71(C2×D131)⋊C4—22no—
72(C2×D131)⋊C4—22no—
73C24×C131C2312yes—
74C24×C131C2311yescharacteristic
75C2×D4×C131—24no—
76C2×D4×C131—24no—
77C2×D4×C131—24no—
78C2×D4×C131—22no—
79C2×D4×C131—42no—
80C22×C4×C131C2311yescharacteristic
81C22×C4×C131C2312yes—
82C22×C4×C131C2311yescharacteristic
83C22⋊C4×C131—22no—
84C22⋊C4×C131—42no—
85C22×D131—21no—
86C22×D131—21no—
87C22×D131—22no—
88C22×D131—28no—
89C22×D131—21no—
90C22×D131—22no—
91C22×D131—21no—
92C22×D131—22no—
93C22×D131—48no—
94C22×D131—48no—
95C22×D131—44no—
96C22×D131—44no—
97C22×D131C2×D414yes—
98C22×D131C2×D412yes—
99C22×D131#4112yescomplemented
100Dic262—28no—
101C22⋊D131—216no—
102C22⋊D131—48no—
103C2×Dic131—216no—
104C2×Dic131—22no—
105C2×Dic131—22no—
106C2×Dic131—22no—
107C2×Dic131—21no—
108C2×Dic131—22no—
109C2×Dic131—22no—
110C2×Dic131—22no—
111C2×Dic131—22no—
112C2×Dic131C2×D412yes—
113C2×Dic131C2×D414yes—
114C4×D131—28no—
115C4×D131—44no—
116C23×C131—21no—
117C23×C131—22no—
118C23×C131—22no—
119C23×C131—22no—
120C23×C131—22no—
121C23×C131—22no—
122C23×C131—21no—
123C23×C131—22no—
124C23×C131—22no—
125C23×C131—22no—
126C23×C131C2×D414yes—
127C23×C131#4112yescomplemented
128C23×C131C2411yescharacteristic
129D4×C131—28no—
130D4×C131—48no—
131C2×C4×C131—22no—
132C2×C4×C131—21no—
133C2×C4×C131—22no—
134C2×C4×C131—22no—
135C2×C4×C131—22no—
136C2×C4×C131—22no—
137C2×C4×C131C2×D412yes—
138C22×C131—24no—
139C22×C131—22no—
140C22×C131—24no—
141C22×C131—24no—
142C22×C131—24no—
143C22×C131—41no—
144C22×C131—42no—
145C22×C131—42no—
146C22×C131—41no—
147C22×C131—42no—
148C22×C131—42no—
149C22×C131—42no—
150C22×C131C22×D411yescharacteristic
151C22×C131C22×D412yes—
152C22×C131C2×#4111yescharacteristic
153C22×C131C2×#4111yescharacteristic
154C22×C131H(5,𝔽2)12yes—
155C2×D131—22no—
156C2×D131—22no—
157C2×D131—24no—
158C2×D131—24no—
159C2×D131—22no—
160C2×D131—24no—
161C2×D131—416no—
162C2×D131—41no—
163C2×D131—41no—
164C2×D131—42no—
165C2×D131#12014yes—
166C4×C131—22no—
167C4×C131—41no—
168C4×C131—42no—
169Dic131—22no—
170Dic131—24no—
171Dic131—41no—
172Dic131—42no—
173C2×C131—42no—
174C2×C131—44no—
175C2×C131C2×#12011yescharacteristic
176C2×C131#52511yescharacteristic
177C2×C131#53511yescharacteristic
178C2×C131#54612yes—
179C2×C131#54612yes—
180D131—24no—
181D131—42no—
182C131#240411yescharacteristic, complemented, Sylow
183#2404—1311nomaximal, Sylow
184C22×#41—1311no—
185C2×#120—1312no—
186C22×C22⋊C4—1311no—
187#393—1311no—
188#393—1311no—
189#392—1311no—
190#392D13111yescharacteristic, complemented
191#386—1312no—
192#386—1312no—
193#386—1312no—
194#381—1311no—
195C25—1311no—
196C2×#41—2628no—
197C2×#41—2624no—
198C22×Q8—1311no—
199C22×D4—1312no—
200C22×D4—1312no—
201C22×D4C2×D13112yes—
202C22×D4C2×D13111yescharacteristic, complemented
203C23×C4—1312no—
204C23×C4—1311no—
205C23×C4—1311no—
206#120—2628no—
207C2×C22⋊C4—2624no—
208C2×C22⋊C4—1311no—
209C2×C22⋊C4—1312no—
210C2×C22⋊C4—1311no—
211C2×C22⋊C4—1311no—
212C2×C22⋊C4—1312no—
213C2×C22⋊C4—1312no—
214C2×C22⋊C4—1312no—
215C2×C22⋊C4C2×D13111yescharacteristic
216C2×C22⋊C4C2×D13112yescomplemented
217#95—1312no—
218#95—1311no—
219#95—1312no—
220#95—1312no—
221#95C2×D13111yescharacteristic, complemented
222C24—2622no—
223C24—2622no—
224C24—2624no—
225C24—2622no—
226C24—5242no—
227C24—1311no—
228C24—1311no—
229C24C22×D13112yes—
230C24C22×D13111yescharacteristic
231#41—26232no—
232C2×Q8—2624no—
233C2×Q8—2622no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—2624no—
240C2×D4—2624no—
241C2×D4—2624no—
242C2×D4—2624no—
243C2×D4—5242no—
244C2×D4—1318no—
245C22×C4—2624no—
246C22×C4—2622no—
247C22×C4—2624no—
248C22×C4—2624no—
249C22×C4—2624no—
250C22×C4—5242no—
251C22×C4—1312no—
252C22×C4—1311no—
253C22×C4—1311no—
254C22×C4—1312no—
255C22×C4—1314no—
256C22×C4C22×D13112yes—
257C22×C4C22×D13111yescharacteristic
258C22×C4C22×D13111yescharacteristic
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—2624no—
262C22⋊C4—2622no—
263C22⋊C4—2624no—
264C22⋊C4—2622no—
265C22⋊C4—2624no—
266C22⋊C4—2622no—
267C22⋊C4—5242no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—21no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—2628no—
279C23—2621no—
280C23—2621no—
281C23—2622no—
282C23—2621no—
283C23—2622no—
284C23—2621no—
285C23—2622no—
286C23—5248no—
287C23—5248no—
288C23—5244no—
289C23—5244no—
290C23—1312no—
291C23—1314no—
292C23—1312no—
293C23D4×D13114yes—
294C23#1452351212yescomplemented
295C23C23×D13111yescharacteristic
296Q8—2628no—
297D4—28no—
298D4—48no—
299D4—26216no—
300D4—5248no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—2628no—
308C2×C4—26216no—
309C2×C4—2622no—
310C2×C4—2622no—
311C2×C4—2622no—
312C2×C4—2621no—
313C2×C4—2622no—
314C2×C4—2622no—
315C2×C4—2622no—
316C2×C4—2622no—
317C2×C4—5244no—
318C2×C4—1312no—
319C2×C4—1314no—
320C2×C4D4×D13112yes—
321C22—24no—
322C22—24no—
323C22—24no—
324C22—24no—
325C22—22no—
326C22—41no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—41no—
332C22—42no—
333C22—2622no—
334C22—2624no—
335C22—2624no—
336C22—2622no—
337C22—2624no—
338C22—2622no—
339C22—52416no—
340C22—5241no—
341C22—5242no—
342C22—5241no—
343C22—1314no—
344C22C2×D4×D13111yescharacteristic
345C22C2×D4×D13112yes—
346C22#41×D13111yescharacteristic
347C22C2×#1452351211yescharacteristic
348C22#1686229112yes—
349C4—22no—
350C4—41no—
351C4—42no—
352C4—2622no—
353C4—2624no—
354C4—5241no—
355C4—5242no—
356C2—42no—
357C2—44no—
358C2—2624no—
359C2—5242no—
360C2#120×D13111yescharacteristic
361C2#2255382211yescharacteristic
362C2#2255387511yescharacteristic
363C2#2255394912yes—
364C2#2255395112yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#392×C131C2
hypercenter#392D131
hypocenterC131#2404
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #392

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#14523512C2semidirect
2. C2×#16862214C2semidirect
3. C2×C22⋊C4×D131C2semidirect
4. #22553225C2semidirect
5. #22553224C2semidirect
6. #22553220C2semidirect
7. #22553192C2semidirect
8. #22553189C2semidirect
9. #22553188C2semidirect
10. #22553174C2semidirect
11. #392×C131C2semidirect
15. C22×Dic262C22semidirect
18. C23×Dic131C22semidirect
24. C2×C22⋊Dic131C22semidirect
30. C131⋊#95C22semidirect
31. C131⋊#95C22semidirect
32. C131⋊#95C22semidirect
33. C131⋊#95C22semidirect
35. C22×D4×C131C22semidirect
37. C2×C22⋊C4×C131C22semidirect
38. #95×C131C22semidirect
54. C2×C22⋊D131D4semidirect
65. C2×C4×D131D4semidirect
99. C22×D131#41semidirect
127. C23×C131#41semidirect
182. C131#2404semidirect
190. #392D131semidirect
202. C22×D4C2×D131semidirect
216. C2×C22⋊C4C2×D131semidirect
221. #95C2×D131semidirect
294. C23#14523512semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145235122
C2×#168622142
C2×C22⋊C4×D1312
#225532252
#225532242
#225532202
#225531922
#225531892
#225531882
#225531742
#392×C1312
#2404131